Polygonal numbers are those that are arranged in shapes as they enlarge. Explore the polygonal numbers drawn here.

While we were sorting some papers we found 3 strange sheets which seemed to come from small books but there were page numbers at the foot of each page. Did the pages come from the same book?

What can you say about these shapes? This problem challenges you to create shapes with different areas and perimeters.

Can you make dice stairs using the rules stated? How do you know you have all the possible stairs?

Use the interactivity to investigate what kinds of triangles can be drawn on peg boards with different numbers of pegs.

How can you arrange these 10 matches in four piles so that when you move one match from three of the piles into the fourth, you end up with the same arrangement?

Can you continue this pattern of triangles and begin to predict how many sticks are used for each new "layer"?

An investigation that gives you the opportunity to make and justify predictions.

Can you dissect an equilateral triangle into 6 smaller ones? What number of smaller equilateral triangles is it NOT possible to dissect a larger equilateral triangle into?

Tom and Ben visited Numberland. Use the maps to work out the number of points each of their routes scores.

How many ways can you find to do up all four buttons on my coat? How about if I had five buttons? Six ...?

These squares have been made from Cuisenaire rods. Can you describe the pattern? What would the next square look like?

We can arrange dots in a similar way to the 5 on a dice and they usually sit quite well into a rectangular shape. How many altogether in this 3 by 5? What happens for other sizes?

Put the numbers 1, 2, 3, 4, 5, 6 into the squares so that the numbers on each circle add up to the same amount. Can you find the rule for giving another set of six numbers?

What happens when you round these numbers to the nearest whole number?

What happens when you round these three-digit numbers to the nearest 100?

Use two dice to generate two numbers with one decimal place. What happens when you round these numbers to the nearest whole number?

This challenge focuses on finding the sum and difference of pairs of two-digit numbers.

Find the sum and difference between a pair of two-digit numbers. Now find the sum and difference between the sum and difference! What happens?

This challenge encourages you to explore dividing a three-digit number by a single-digit number.

In how many different ways can you break up a stick of 7 interlocking cubes? Now try with a stick of 8 cubes and a stick of 6 cubes.

In each of the pictures the invitation is for you to: Count what you see. Identify how you think the pattern would continue.

Sweets are given out to party-goers in a particular way. Investigate the total number of sweets received by people sitting in different positions.

Only one side of a two-slice toaster is working. What is the quickest way to toast both sides of three slices of bread?

Compare the numbers of particular tiles in one or all of these three designs, inspired by the floor tiles of a church in Cambridge.

This task follows on from Build it Up and takes the ideas into three dimensions!

This challenge, written for the Young Mathematicians' Award, invites you to explore 'centred squares'.

Can you find all the ways to get 15 at the top of this triangle of numbers? Many opportunities to work in different ways.

How many different journeys could you make if you were going to visit four stations in this network? How about if there were five stations? Can you predict the number of journeys for seven stations?

What happens if you join every second point on this circle? How about every third point? Try with different steps and see if you can predict what will happen.

Take a counter and surround it by a ring of other counters that MUST touch two others. How many are needed?

In a Magic Square all the rows, columns and diagonals add to the 'Magic Constant'. How would you change the magic constant of this square?

How many centimetres of rope will I need to make another mat just like the one I have here?

Can you find a way of counting the spheres in these arrangements?

Think of a number, square it and subtract your starting number. Is the number youâ€™re left with odd or even? How do the images help to explain this?

Can you put the numbers 1-5 in the V shape so that both 'arms' have the same total?

Here are two kinds of spirals for you to explore. What do you notice?

Here are some arrangements of circles. How many circles would I need to make the next size up for each? Can you create your own arrangement and investigate the number of circles it needs?

Try out this number trick. What happens with different starting numbers? What do you notice?

Can you see why 2 by 2 could be 5? Can you predict what 2 by 10 will be?

Place the numbers from 1 to 9 in the squares below so that the difference between joined squares is odd. How many different ways can you do this?

Imagine an infinitely large sheet of square dotty paper on which you can draw triangles of any size you wish (providing each vertex is on a dot). What areas is it/is it not possible to draw?

Delight your friends with this cunning trick! Can you explain how it works?

Watch this animation. What do you notice? What happens when you try more or fewer cubes in a bundle?

The Tower of Hanoi is an ancient mathematical challenge. Working on the building blocks may help you to explain the patterns you notice.

Watch this animation. What do you see? Can you explain why this happens?

Find the sum of all three-digit numbers each of whose digits is odd.

Try adding together the dates of all the days in one week. Now multiply the first date by 7 and add 21. Can you explain what happens?

In this problem we are looking at sets of parallel sticks that cross each other. What is the least number of crossings you can make? And the greatest?

Can you work out how to win this game of Nim? Does it matter if you go first or second?